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1 Mah-NeRu All Russian mahemaical poral Roman Popovych, On elemens of high order in general finie fields, Algebra Discree Mah, 204, Volume 8, Issue 2, Use of he all-russian mahemaical poral Mah-NeRu implies ha you have read and agreed o hese erms of use hp://wwwmahneru/eng/agreemen Download deails: IP: November 8, 208, 07:7:37

2 Algebra and Discree Mahemaics Volume 8 204) Number 2, pp Journal Algebra and Discree Mahemaics RESEARCH ARTICLE On elemens of high order in general finie fields Roman Popovych Communicaed by I P Shesakov Absrac We show ha he Gao s consrucion gives for any finie field F q n elemens wih he muliplicaive order a leas ) d, where d = 2 log i q n, = log d n n+ Inroducion I is well known ha he muliplicaive group of a finie field is cyclic A generaor of he group is called a primiive elemen The problem of consrucing efficienly a primiive elemen for a given finie field is nooriously difficul in he compuaional heory of finie fields Tha is why one considers less resricive quesion: o find an elemen wih high muliplicaive order We are no required o compue he exac order of he elemen I is sufficien in his case o obain a lower bound on he order High order elemens are needed in several applicaions Such applicaions include bu are no limied o crypography, coding heory, pseudo random number generaion and combinaorics Throughou his paper F q is a field of q elemens, where q is a power of prime number p We use F q o denoe he muliplicaive group of F q Previous work If no consrain is pu on he exension degree n, very few resuls are known Gao gives in [5] an algorihm for consrucing high order elemens for general exensions F q n of finie field F q wih log q n 4 log lower bound on he order n q 2 log q n) 2 His algorihm assumes some 200 MSC: T30 Key words and phrases: finie field, muliplicaive order, Diophanine inequaliy

3 296 On elemens of high order in general finie fields reasonable bu unproved conjecure Conflii [4] provided a more careful analysis of resuls from [5] A polynomial algorihm ha find a primiive elemen in finie field of small characerisic is described in [8] However, he algorihm relies on wo unproved assumpions, and he second assumpion is no suppored by any compuaional example For special finie fields, i is possible o consruc elemens which can be proved o have much higher orders Exensions conneced wih a noion of Gauss period are considered in [,6,7,0] The lower bound on he order equals o exp25 n 2) 3n 2) Exensions based on he Kummer and Arin-Schreier polynomials are considered in [2, ] Some generalizaion of he exensions is given in [3] Field exension based on he Kummer polynomial is of he form F q [x]/x n a) I is shown in [2] how o consruc high order elemen in he exension F q [x]/x n a) wih he condiion q mod n) The lower bound 5, 8 n is obained in his case High order elemens are consruced in [] for Kummer exensions wihou he condiion q mod n) wih lower bound 2 3 2n Voloch [2, 3] proposed a mehod which consrucs an elemen of order a leas explog n) 2 ) in finie fields from ellipic curves Our resuls Se F q θ) = F q n = F q [x]/fx), where fx) is an irreducible polynomial over F q of degree n and θ = x mod fx) is he cose of x We improve he Gao s consrucion and is modificaion by Conflii for any finie field F q n The mehod similar o ha in [4, 5] is used for he proof Our main resul is he following heorem Theorem Se d = 2 log q n, = log d n The θ has in he field F q θ) = F q n = F q [x]/fx) he muliplicaive order a leas Preliminaries n + ) d i ) We recall ha he muliplicaive order ordβ) of he elemen β F q n is he smalles posiive ineger u such ha β u = Le m be he smalles power of q greaer or equal o n The Gao approach [5] depends on he following conjecure

4 R Popovych 297 Conjecure For any ineger n, here exis a polynomial gx) F q [x] of degree d a mos 2 log q n such ha x m gx) has irreducible facor fx) of degree n If he { conjecure holds, hen clearly θ m = gθ) Gao considered he } se S = u im i 0 u i µ and chase and µ from he condiion µd < n He proved ha θ u are disinc elemens for u S, ook =, µ = log q n n and showed S = µ + ) 4 log n q 2 log q n) 2 logq n 2 log q d Conflii [4] considered he following se S = { u i m i 0 u i µ i, n d i µ i n } d i and chase and µ from he condiion µ id i < n He proved ha θ u are disinc elemens for u S, ook = log d n and showed S = µ i + ) n ) Subsiuing = log d n ino 2), we obain ordθ) ) nd 2 log d n log 2 d n d i 2) The resuls from [4, 5] are based on he following saemen see [5, Theorem 4]) Lemma Suppose ha fx) F q [x] is no a monomial nor a binomial of he form ax pl +b, where p is he characerisic of F q Then he polynomials f ) x) = fx), f k) x) = f k ) x), k 2 are muliplicaively independen in F q [x], ha is, if f ) x)) k f 2) x)) k 2 f s) x)) ks = for any inegers s, k,,k s, hen k = k 2 = = k s = 0 The following lemma [9] gives lower bound for he number of nonnegaive soluions of linear Diophanine inequaliy

5 298 On elemens of high order in general finie fields Lemma 2 Le a 0,, a r be posiive inegers wih gcda 0,, a r )= Then he number of non-negaive ineger soluions x 0,, x r of he linear Diophanine inequaliy is a leas r m + r r a i x i m, ) r a i 2 Main resul To improve he Conflii resul we consider he se of soluions u 0,, u r of he linear Diophanine inequaliy r d i u i m, and show ha θ u are disinc elemens in F q n for all u S We give below he proof of our main resul Proof of Theorem If θ is a roo of x m gx), hen since m is a power of q, applying ieraively he Frobenius auomorphism we have θ mi = g i) θ), i N 3) where as in he saemen of lemma, g i) x) is he polynomial obained by composing gx) wih iself i imes Consider he se S = { u i m i d i u i n, u i 0 For every elemen u S we consruc he power θ u ha belongs o he group generaed by θ We show ha if wo elemens u, v S are disinc, hen he corresponden powers do no coincide Assume ha elemens u = u im i and v = v im i from S are disinc, and he corresponden powers are equal: θ u = θ v Then we have θ mi) u i = θ mi) v i }

6 R Popovych 299 Taking ino accoun he equaliy 3), we ge g i) θ)) ui = g i) θ)) vi Define he following polynomials h x) = u i >v i g i) θ)) ui v i and h 2 x) = v i >u i g θ)) i) vi u i Then h θ) = h 2 θ), and since gx) is he characerisic polynomial of θ, we wrie: h x) = h 2 x) mod fx) As g i) x) has degree d i, h x) is of degree a mos u id i n and h 2 x) is of degree a mos v id i n Thus h x) and h 2 x) mus be equal as polynomials over F q Therefore g i) x)) ui v i = According o lemma he polynomials g i) x) are muliplicaively independen in F q [x] So u i = v i for i = 0,,, and hus u = v - a conradicion Hence, he number of elemens of S and he muliplicaive order of θ) is a leas he number of nonnegaive ineger soluions of he Diophanine inequaliy di x i n Finally, applying lemma 2, we have and he resul follows S n + ) d i, Now we compare our resul wih he Conflii resul Le us calculae for his purpose he raio R of he bound ) o he bound 2): R = i= n + i n I is clear ha R > for any q and n recall ha depends on q and n) We provide below a few numerical examples of lower bounds on he muliplicaive orders of he considered previously elemen θ Denoe lower bounds on he orders of θ obained in [4] and in his paper by b and b 2 respecively Values of q, n, d,, b, b 2 and R in examples -3 are given in he able i

7 300 On elemens of high order in general finie fields No q n d b b 2 R , , , , 6 0, , , , ,52 References [] O Ahmadi, I E Shparlinski, J F Voloch, Muliplicaive order of Gauss periods, In J Number Theory, Vol6 200), No4, pp [2] Q Cheng, On he consrucion of finie field elemens of large order, Finie Fields Appl, Vol 2005), No3, pp [3] Q Cheng, S Gao, D Wan, Consrucing high order elemens hrough subspace polynomials, Discree algorihms: Proc 23rd ACM-SIAM Symp Kyoo, Japan, 7 9 January 202) Omnipress, Philadelphia, USA, 20, pp [4] A Conflii, On elemens of high order in finie fields, Crypography and compuaional number heory: Proc Workshop Singapore, November 999) Birkhauser, Basel, 200, pp 4 [5] S Gao, Elemens of provable high orders in finie fields, Proc Amer Mah Soc, Vol27 999), No6, pp [6] J von zur Gahen, IE Shparlinski, Orders of Gauss periods in finie fields, Appl Algebra Engrg Comm Compu, Vol9 998), No, pp5-24 [7] J von zur Gahen, IE Shparlinski, Gauss periods in finie fields, Finie Fields and heir Applicaions: Proc 5h Conf Ausburg, Germany, 2 6 Augus 999) Springer, Berlin, 200, pp62-77 [8] M-DHuang, A K Narayanan, Finding primiive elemens in finie fields of small characerisic, arxiv , 203 [9] T A Lambe, Bounds on he Number of Feasible Soluions o a Knapsack Problem, SIAM J Applied Mah, Vol26 974), No2, pp [0] R Popovych, Elemens of high order in finie fields of he form F q[x]/φ rx), Finie Fields Appl, Vol8 202), No4, pp [] R Popovych, Elemens of high order in finie fields of he form F q[x]/x m a), Finie Fields Appl, Vol9 203), No, pp86-92 [2] JF Voloch, On he order of poins on curves over finie fields, Inegers, Vol7 2007), A49 [3] JF Voloch, Elemens of high order on finie fields from ellipic curves, Bull Ausral Mah Soc, Vol8 200), No3, pp Conac informaion R Popovych Lviv Polyechnic Naional Universiy, Insiue of Compuer Technologies, Bandery Sr, 2, Lviv, 7903, Ukraine s): rombp07@gmailcom Received by he ediors: and in final form